How would you find the equations of the asymptotes

[chaoseverlasting]

$$L_1L_2 +\lambda =0$$
$$S+\lambda =0$$ (Such that D=0)
$$S+2\lambda =0$$

In case of a hyperbola, S is the pair of straight lines representing the asymptotes and $$\lambda$$ is any parameter.

My question is, are the first two equations the same? How would you find the equations of the asymptotes if you were given the equation of the curve.

The third equation is the conjugate hyperbola if $$S+\lambda =0$$ represents the original hyperbola. Is there any other way to find the conjugate hyperbola?

If $$\frac{x^2}{a^2} - \frac{y^2}{b^2} =1$$ is the equation of the original hyperbola, then does the equation $$\frac{x^2}{a^2} - \frac{y^2}{b^2} =-1$$ represent the conjugate hyperbola?

 

[HallsofIvy]

I can make very little sense of this. What are L₁ and L₂? Are they linear functions? What is S?

 

[chaoseverlasting]

$$L_1 and L_2$$ are linear functions. S represents a pair of straight lines.

 

[drpizza]

Yes, to your very last part. I found the rest to be confusing.